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Applications of Vector Algebra Model Question Paper

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 40
    5 x 1 = 5
  1. If a vector \(\vec { \alpha } \) lies in the plane of \(\vec { \beta } \) and \(\vec { \gamma } \) , then

    (a)

    \([\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]\) = 1

    (b)

    \([\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]\)= -1

    (c)

    \([\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]\)= 0

    (d)

    \([\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]\)= 2

  2. If \(\vec { a } \) and \(\vec { b } \) are unit vectors such that \([\vec { a } ,\vec { b },\vec { a } \times \vec { b } ]=\frac { \pi }{ 4 } \), then the angle between \(\vec { a } \) and \(\vec { b } \) is

    (a)

    \(\frac { \pi }{ 6 } \)

    (b)

    \(\frac { \pi }{ 4 } \)

    (c)

    \(\frac { \pi }{ 3 } \)

    (d)

    \(\frac { \pi }{ 2 } \)

  3. If \(\vec { a } ,\vec { b } ,\vec { c } \) are three non-coplanar vectors such that \(\vec { a } \times (\vec { b } \times \vec { c } )=\frac { \vec { b } +\vec { c } }{ \sqrt { 2 } } \), then the angle between

    (a)

    \(\frac { \pi }{ 2 } \)

    (b)

    \(\frac { 3\pi }{ 6 } \)

    (c)

    \(\frac { \pi }{ 4 } \)

    (d)

    \( { \pi }\)

  4. The number of vectors of unit length perpendicular to the vectors \(\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) \) and \(\left( \overset { \wedge }{ j } +\overset { \wedge }{ k } \right) \)is

    (a)

    1

    (b)

    2

    (c)

    3

    (d)

  5. If \(\overset { \rightarrow }{ d } =\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) +\overset { \rightarrow }{ b } \times \left( \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } \right) +\overset { \rightarrow }{ c } \times \left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) \), then

    (a)

    \(\left| \overset { \rightarrow }{ d } \right| \)

    (b)

    \(\overset { \rightarrow }{ d } =\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \)

    (c)

    \(\overset { \rightarrow }{ d } =\overset { \rightarrow }{ 0 } \)

    (d)

    a, b, c are coplanar

  6. 5 x 2 = 10
  7. If \(\vec { a } =\hat { i } -2\hat { j } +3\hat { k }, \vec { 6 } =2\hat { i } +\hat { j } -2\hat { k }, \vec { c } =3\hat { i } +2\hat { j } +\hat { k } \) find \(\vec { a } .(\vec { b } \times \vec { c } )\).

  8. Find the volume of the parallelepiped whose coterminous edges are represented by the vectors \(-6\hat { i } +14\hat { j } +10\hat { k } ,14\hat { i } -10\hat { j } -6\hat { k } \) and \(2\hat { i } +4\hat { j } -2\hat { k } \)

  9. Find the Cartesian equation of a.line passing through the pointsA(2, -1, 3) and B(4, 2, 1)

    ()

    -1

  10. Find the parametric form of vector equation of the plane passing through the point (1, -1, 2) having 2, 3, 3.as direction ratios of normal to the plane.

    ()

    2

  11. If the planes \({ \overset { \rightarrow }{ r } }.\left( \overset { \wedge }{ i } +2\overset { \wedge }{ j } +3\overset { \wedge }{ k } \right) =7\)=and \({ \overset { \rightarrow }{ r } }.\left( \lambda \overset { \wedge }{ i } +2\overset { \wedge }{ j } -7\overset { \wedge }{ k } \right) =26\) are perpendicular. Find the value of λ.

    ()

    ∆=4

  12. 5 x 3 = 15
  13. In triangle, ABC the points, D, E, F are the midpoints of the sides, BC, CA and AB respectively. Using vector method, show that the area of ΔDEF is equal to \(\frac{1}{2}\)(area of ΔABC )

  14. Prove by vector method that the diagonals of a rhombus bisect each other at right angles.

  15. Find the equation of the plane through the intersection of the planes lx-3y+ z-4 -0 and x - y + Z + 1 - 0 and perpendicular to the plane x + 2y - 3z + 6 = 0

    ()

    \(\overset { \rightarrow }{ a } \)\(\overset { \rightarrow }{ c } \)

  16. Prove that \(\left[ \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } \right] \)=\(\left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] \)

    ()

    0

  17. Show that the four points whose position vectors are \(6\overset { \wedge }{ i } -7\overset { \wedge }{ j } ,16\overset { \wedge }{ i } -29\overset { \wedge }{ j } -4\overset { \wedge }{ k } ,3\overset { \wedge }{ i } -6\overset { \wedge }{ j } \) are co-planar

    ()

    1

  18. 2 x 5 = 10
  19. Prove by vector method that sin(α + β )=sin α cos β + cos α sin β

  20. If \(\vec { a } =-2\hat { i } +3\hat { j } -2\hat { k } ,\vec { b } =3\hat { i } -\hat { j } +3\hat { k } ,\vec { c } =2\hat { i } -5\hat { j } +\hat { k } \) find \((\vec { a } \times \vec { b } )\times \vec { c } \) and \((\vec { a } \times \vec { b } )\times \vec { c } \). State whether they are equal.

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